A blow-up result for the semilinear Euler-Poisson-Darboux-Tricomi equation with critical power nonlinearity

TL;DR

This paper establishes a blow-up result for a generalized semilinear Euler-Poisson-Darboux equation with critical power nonlinearity, extending classical wave equation results to equations with polynomially growing propagation speed.

Contribution

It introduces a novel comparison argument and integral representation approach to prove blow-up and lifespan estimates for this class of equations, which was not previously addressed.

Findings

Proves blow-up for the equation with critical power nonlinearity.

Derives sharp lifespan upper bounds matching classical wave equation results.

Extends blow-up analysis to equations with polynomially growing propagation speed.

Abstract

In this paper, we prove a blow-up result for a generalized semilinear Euler-Poisson-Darboux equation with polynomially growing speed of propagation, when the power of the semilinear term is a shift of the Strauss' exponent for the classical semilinear wave equation. Our proof is based on a comparison argument of Kato-type for a second-order ODE with time-dependent coefficients, an integral representation formula by Yagdjian and the Radon transform. As byproduct of our method, we derive upper bound estimates for the lifespan which coincide with the sharp one for the classical semilinear wave equation in the critical case.